# Thread: Prove the Bonferroni inequality

1. ## Prove the Bonferroni inequality

Hi,

I would like to know how to prove the Bonferroni inequatlity (I have actually started but at some point, I don't know how to prove).

1- P(A1) - P(A2)- P(A3)

Note:

Proof:

= 1 - P(A1 U A2 U A3)
= 1 - P(A1) - P(A2) - P(A3) + P(A1 n A2)
+ P(A1 n A3) + P(A2 n A3)
- P(A1 n A2 n A3) (1)
1- P(A1) - P(A2)- P(A3) (2)

I would like to know how to justify, proof that
P(A1 n A2) + P(A1 n A3) + P(A2 n A3) - P(A1 n A2 n A3) is positive?
Like how do I go from the step before last step to my last step, that is from (1) to (2)?
Thank you

2. This follows from

$\displaystyle P(\cup A_n)\le \sum P(A_n)$

If you wish to see why 'P(A1 n A2) + P(A1 n A3) + P(A2 n A3) - P(A1 n A2 n A3) is positive'
draw a Venn diagram http://www.learnnc.org/reference/Venn%20diagram
The triple intersection is contained repeatedly in those other probabilities.

Note that if $\displaystyle A\subset B$ then $\displaystyle P(A)\le P(B)$.

This comes from $\displaystyle P(B)=P(AB)+P(A'B)\ge P(AB)=P(A)$.

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### bonferroni inequality proof

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